Fractional Poisson random sum and its associated normal variance mixture

Abstract

In this work, we study the partial sums of independent and identically distributed random variables with the number of terms following a fractional Poisson (FP) distribution. The FP sum contains the Poisson and geometric summations as particular cases. We show that the weak limit of the FP summation, when properly normalized, is a mixture between the normal and Mittag-Leffler distributions, which we call by Normal-Mittag-Leffler (NML) law. A parameter estimation procedure for the NML distribution is developed and the associated asymptotic distribution is derived. Simulations are run to check the performance of the proposed estimators under finite samples. An empirical illustration on the daily log-returns of the Brazilian stock exchange index (IBOVESPA) shows that the NML distribution captures better the tails than some of its competitors. Related problems such as a mixed Poisson representation for the FP law and the weak convergence for the Conway-Maxwell-Poisson random sum are also addressed.

Keywords:

• Estimation
• fractional Poisson distribution
• log-returns
• Mittag-Leffler distribution
• weak convergence

2020 Mathematical Subject Classification:

• 60F05
• 62E20
• 62Fxx

Acknowledgements

We thank the three anonymous referees for their comments and suggestions. G. Oliveira thanks the partial financial support from Coordenação de Aperfeiçoamento de Pessoal de Nível Superior (CAPES-Brazil). W. Barreto-Souza acknowledges support for his research from the KAUST Research Fund and NIH 1R01EB028753-01.